Eulerian contributions to the particle velocity in Stokes and Gerstner waves

For inviscid periodic wave motion, we derive a novel expression in Lagrangian variables for the Stokes drift, which is valid for rotational as well as irrotational waves. The derivation confirms that the Lagrangian mean velocity can be expressed as the sum of the Eulerian mean velocity and the Stokes drift. However, from the Stokes drift part of this expression, we find that the rotational Gerstner wave has a non-zero Stokes drift. Since the Lagrangian mean velocity is zero for this particular wave, we obviously must have a non-zero Eulerian mean velocity in this case, cancelling the Stokes drift. To discuss this problem in detail, we return to the basic kinematics of periodic wave motion in fluids. We avoid time averaging and consider the classic problem of how the Lagrangian particle velocity develops in time, resulting in a Eulerian velocity (expressed in Lagrangian variables) plus the Stokes velocity. We discuss the implication for irrotational deep-water Stokes waves and rotational Gerstner waves. It is demonstrated that the Eulerian velocity, expressed in Lagrangian variables, is different for the two wave types. This explains why the Lagrangian mean velocity in the Stokes wave is equal to the Stokes drift, while it is zero for the Gerstner wave.